Finite Group Databasebeta

← #29671285

Subgroups — #29671285

Order 15616.

778 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457632C211yesmaximal, characteristic, complemented
2#21457626C211yesmaximal, characteristic, complemented
3#21457420C211yesmaximal, characteristic, complemented
4#21445409C211yesmaximal, characteristic, complemented
5#21445284C211yesmaximal, characteristic, complemented
6#21445281C211yesmaximal, characteristic, complemented
7#21445258C211yesmaximal, characteristic, complemented
8#21445254C211yesmaximal, characteristic, complemented
9#21445054C211yesmaximal, characteristic, complemented
10#21445052C211yesmaximal, characteristic, complemented
11#21444873C211yesmaximal, characteristic, complemented
12#21444872C211yesmaximal, characteristic, complemented
13#21444702C211yesmaximal, characteristic, complemented
14D8×#92418C211yesmaximal, characteristic, complemented
15C61×#3272C211yesmaximal, characteristic, complemented
16C2×#14519801—21no—
17#16840416C2211yescharacteristic
18D4×D244C2211yescharacteristic, complemented
19#16840165C2211yescharacteristic, complemented
20#16840138C2211yescharacteristic
21#16840135C2211yescharacteristic, complemented
22#16840110C2211yescharacteristic, complemented
23#16840108C2211yescharacteristic, complemented
24#16840075—21no—
25#16840073—21no—
26#16840042C2211yescharacteristic, complemented
27#16840039C2211yescharacteristic, complemented
28#16840009—21no—
29#16839996—21no—
30#16839996—21no—
31#16839967C2211yescharacteristic, complemented
32#16839957C2211yescharacteristic
33#16839957C2211yescharacteristic
34#16839954C2211yescharacteristic
35#16839914—21no—
36#16839908—21no—
37#16839896—21no—
38#16839895—21no—
39C4×#11758199C2211yescharacteristic
40#16839841C2211yescharacteristic, complemented
41D4×#92418C2211yescharacteristic, complemented
42D4×#92418C2211yescharacteristic, complemented
43C2×#14519698—21no—
44C2×#14519697—21no—
45#16839515C2211yescharacteristic, complemented
46#16839514C2211yescharacteristic, complemented
47#16839509C2211yescharacteristic, complemented
48#16839508C2211yescharacteristic, complemented
49#16839506C2211yescharacteristic, complemented
50#16839505C2211yescharacteristic, complemented
51D4⋊#92418C2211yescharacteristic, complemented
52D4⋊#92418C2211yescharacteristic, complemented
53D4⋊#92418C2211yescharacteristic, complemented
54#16839489C2211yescharacteristic, complemented
55C8×#92418C2211yescharacteristic, complemented
56C2×AGL(1,ℤ/8)×C61—21no—
57C61×#547C2211yescharacteristic
58D42×C61C2211yescharacteristic, complemented
59C61×#492C2211yescharacteristic, complemented
60C61×#468—21no—
61C61×#461C2211yescharacteristic, complemented
62C61×#459C2211yescharacteristic
63C61×#450—21no—
64C61×#449—21no—
65C4×D8×C61C2211yescharacteristic
66C61×#436C2211yescharacteristic, complemented
67C2×#11758224—21no—
68C22×D244—21no—
69C2×D4×D61—21no—
70#14519801—44no—
71C2×#11758190—21no—
72C2×#11758190—21no—
73C2×#11758200—21no—
74C2×#11758199—21no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77C2×#11758199—41no—
78C2×#11758199D411yescharacteristic
79C2×#11758189—21no—
80C22×#92418—21no—
81C22×#92418—21no—
82#14519777C2311yescharacteristic
83#14519774C2311yescharacteristic
84#14519761—41no—
85#14519757—21no—
86#14519752—21no—
87#14519751—21no—
88#14519751—21no—
89#14519747—41no—
90C4×D244C2311yescharacteristic
91C2×C4⋊Dic61—21no—
92#14519710—21no—
93#14519710—21no—
94#14519710—41no—
95#14519710D411yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519700—21no—
100#14519698—21no—
101#14519698—21no—
102#14519698—21no—
103#14519698—21no—
104#14519698D411yescharacteristic
105#14519697—21no—
106#14519697—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112C22⋊#92418—21no—
113#14519689—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C22×D4×C61—21no—
118C22×D4×C61—21no—
119AGL(1,ℤ/8)×C61—44no—
120C2×SD16×C61—21no—
121C2×SD16×C61—21no—
122C2×SD16×C61—21no—
123C2×D8×C61—21no—
124C2×D8×C61D411yescharacteristic
125C2×M16×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127#127×C61C2311yescharacteristic
128#123×C61—41no—
129C22⋊Q8×C61—21no—
130#121×C61—21no—
131#121×C61—21no—
132#121×C61—21no—
133#120×C61—41no—
134C4×D4×C61C2311yescharacteristic
135C4×D4×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137C2×C4⋊C4×C61—21no—
138#107×C61D411yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#102×C61—21no—
142#102×C61D411yescharacteristic
143#102×C61D411yescharacteristic
144C22⋊C8×C61—21no—
145C4×C8×C61C2311yescharacteristic
146C24×C61—41no—
147C23×D61—41no—
148#41×C61—42no—
149#41×C61—81no—
150C2×Q8×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61—41no—
156C2×D4×C61—42no—
157C2×D4×C61—42no—
158C2×D4×C61—41no—
159C2×D4×C61—81no—
160C2×D4×C61—81no—
161C2×D4×C61C2×D411yescharacteristic
162C2×D4×C61C2×D411yescharacteristic
163C2×D4×C61C2×D411yescharacteristic
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—41no—
168C2×C22⋊D61—41no—
169C2×C22⋊D61—41no—
170C22×Dic61—41no—
171#11758224—42no—
172#11758224—81no—
173D4×D61—42no—
174D4×D61—81no—
175C2×D244—21no—
176C2×D244—21no—
177C2×D244—21no—
178C2×D244—42no—
179C2×D244—41no—
180C2×D244—81no—
181C2×D244C2×D411yescharacteristic
182C2×C4×D61—21no—
183C2×Dic122—21no—
184SD16×C61—42no—
185SD16×C61—42no—
186SD16×C61—81no—
187D8×C61—42no—
188D8×C61D814yescomplemented
189M16×C61—42no—
190C2×C8×C61—21no—
191C2×C8×C61C2×D411yescharacteristic
192C2×C8×C61C2×D411yescharacteristic
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—41no—
196C4⋊C4×C61C2×D411yescharacteristic
197C4⋊C4×C61C2×D411yescharacteristic
198C4⋊C4×C61C2×D411yescharacteristic
199C22⋊C4×C61—21no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—21no—
202C22⋊C4×C61—41no—
203C22⋊C4×C61—41no—
204C42×C61C2411yescharacteristic
205#11758200—42no—
206#11758199—41no—
207#11758199—42no—
208#11758199—82no—
209#11758199—81no—
210#11758199—81no—
211#11758198—21no—
212#11758198—41no—
213#11758198—41no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—41no—
217C4⋊Dic61C2×D411yescharacteristic
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—81no—
223C2×#92418—81no—
224C2×#92418C2×D411yescharacteristic
225C2×#92418C2×D411yescharacteristic
226#11758190—42no—
227#11758190—81no—
228#11758189—42no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—41no—
233C23×C61—82no—
234C23×C61—82no—
235C23×C61—161no—
236C22×D61—21no—
237C22×D61—41no—
238C22×D61—82no—
239C22×D61—82no—
240C22×D61—161no—
241Q8×C61—41no—
242Q8×C61—41no—
243D4×C61—21no—
244D4×C61—21no—
245D4×C61—21no—
246D4×C61—21no—
247D4×C61—41no—
248D4×C61—41no—
249D4×C61—41no—
250D4×C61—41no—
251D4×C61—81no—
252D4×C61—82no—
253D4×C61—82no—
254D4×C61—82no—
255D4×C61C2×D812yes—
256D4×C61C2×D812yes—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—82no—
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C22⋊D61—82no—
272C22⋊D61—82no—
273C2×Dic61—21no—
274C2×Dic61—41no—
275C2×Dic61—82no—
276D244—21no—
277D244—21no—
278D244—41no—
279D244—41no—
280D244—41no—
281D244—41no—
282D244—81no—
283D244—82no—
284C4×D61—41no—
285C4×D61—41no—
286Dic122—41no—
287Dic122—41no—
288C8×C61—21no—
289C8×C61—41no—
290C8×C61C2×D812yes—
291#92418—22no—
292#92418—21no—
293#92418—21no—
294#92418—41no—
295#92418—41no—
296#92418—41no—
297#19884—611nomaximal, Sylow
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—164no—
311C22×C61D4211yescharacteristic
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C2×D61—164no—
319C4×C61—21no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61—41no—
323C4×C61—81no—
324C4×C61C22×D811yescharacteristic
325C4×C61#57511yescharacteristic
326C4×C61#57611yescharacteristic
327C4×C61#57611yescharacteristic
328Dic61—41no—
329Dic61—81no—
330#3294—611no—
331#3294—611no—
332#3272—611no—
333#3272D6111yescharacteristic, complemented
334#1706—611no—
335#1685—611no—
336#1685—611no—
337#1681—611no—
338#1681—611no—
339#1647—611no—
340#1647—611no—
341#1619—611no—
342#1619—611no—
343#1589—611no—
344C8×D8—611no—
345C2×C61—42no—
346C2×C61—42no—
347C2×C61—81no—
348C2×C61—161no—
349C2×C61#327311yescharacteristic
350C2×C61D4×D811yescharacteristic
351C2×C61#329111yescharacteristic
352D61—81no—
353D61—161no—
354C2×AGL(1,ℤ/8)—21no—
355C2×AGL(1,ℤ/8)—1221no—
356#547—611no—
357#547C2×D6111yescharacteristic
358D42—611no—
359D42C2×D6111yescharacteristic, complemented
360#500—611no—
361#493—611no—
362#492—611no—
363#492C2×D6111yescharacteristic, complemented
364#486—611no—
365#486—611no—
366#480—1221no—
367#480—1221no—
368#474—611no—
369#474—611no—
370#468—21no—
371#468—1221no—
372#466—1221no—
373#466—1221no—
374#461—611no—
375#461C2×D6111yescharacteristic, complemented
376#459—611no—
377#459—611no—
378#459—611no—
379#459C2×D6111yescharacteristic
380#450—21no—
381#450—1221no—
382#449—21no—
383#449—1221no—
384#447—1221no—
385#447—1221no—
386C4×D8—611no—
387C4×D8C2×D6111yescharacteristic
388#436—611no—
389#436C2×D6111yescharacteristic, complemented
390C8×D4—611no—
391C8×D4—611no—
392C2×#102—1221no—
393C2×#102—1221no—
394#335—611no—
395#335—611no—
396#332—611no—
397#332—611no—
398#331—611no—
399#331—611no—
400D4⋊C8—611no—
401D4⋊C8—611no—
402D4⋊C8—611no—
403D4⋊C8—611no—
404C82—611no—
405C61#1988411yescharacteristic, complemented, Sylow
406C2×#41—21no—
407C2×#41—1221no—
408C22×D4—21no—
409C22×D4—21no—
410C22×D4—1221no—
411C22×D4—1221no—
412AGL(1,ℤ/8)—44no—
413AGL(1,ℤ/8)—2444no—
414C2×SD16—21no—
415C2×SD16—21no—
416C2×SD16—21no—
417C2×SD16—1221no—
418C2×SD16—1221no—
419C2×SD16—1221no—
420C2×D8—21no—
421C2×D8—611no—
422C2×D8—1221no—
423C2×D8—1221no—
424C2×D8—1221no—
425C2×D8—2441no—
426C2×D8C22⋊D6111yescharacteristic
427C2×M16—21no—
428C2×M16—1221no—
429C22×C8—1221no—
430C22×C8—1221no—
431C4⋊Q8—611no—
432C4⋊Q8C22×D6111yescharacteristic
433#127—611no—
434#127C22×D6111yescharacteristic
435#123—41no—
436#123—2441no—
437C22⋊Q8—21no—
438C22⋊Q8—1221no—
439#121—21no—
440#121—21no—
441#121—21no—
442#121—1221no—
443#121—1221no—
444#121—1221no—
445#120—41no—
446#120—2441no—
447C4×D4—611no—
448C4×D4C22×D6111yescharacteristic
449C4×D4C22×D6111yescharacteristic
450C4×D4C22×D6111yescharacteristic
451C2×C4⋊C4—21no—
452C2×C4⋊C4—1221no—
453#107—611no—
454#107—1221no—
455#107—1221no—
456#107—2441no—
457#107C22⋊D6111yescharacteristic
458C4⋊C8—611no—
459C4⋊C8—611no—
460C4⋊C8—611no—
461C4⋊C8C22×D6111yescharacteristic
462#103—21no—
463#103—1221no—
464#102—21no—
465#102—611no—
466#102—611no—
467#102—1221no—
468#102—1221no—
469#102—1221no—
470#102—1221no—
471#102—1221no—
472#102—1221no—
473#102—1221no—
474#102—1221no—
475#102—1221no—
476#102C22⋊D6111yescharacteristic
477#102C22⋊D6111yescharacteristic
478C22⋊C8—21no—
479C22⋊C8—1221no—
480C22⋊C8—1221no—
481C22⋊C8—1221no—
482C4×C8—611no—
483C4×C8—611no—
484C4×C8C22×D6111yescharacteristic
485C24—41no—
486C24—2441no—
487#41—42no—
488#41—81no—
489#41—4881no—
490#41—2442no—
491C2×Q8—21no—
492C2×Q8—1221no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—41no—
497C2×D4—41no—
498C2×D4—42no—
499C2×D4—42no—
500C2×D4—41no—
501C2×D4—81no—
502C2×D4—81no—
503C2×D4—611no—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—1221no—
507C2×D4—4881no—
508C2×D4—4881no—
509C2×D4—2441no—
510C2×D4—2442no—
511C2×D4—2441no—
512C2×D4—2441no—
513C2×D4—2442no—
514C2×D4D4×D6111yescharacteristic
515C2×D4C2×C22⋊D6111yescharacteristic
516C2×D4C2×C22⋊D6111yescharacteristic
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—41no—
521C22×C4—1221no—
522C22×C4—2441no—
523SD16—42no—
524SD16—42no—
525SD16—81no—
526SD16—4881no—
527SD16—2442no—
528SD16—2442no—
529D8—42no—
530D8—4882no—
531D8—4881no—
532D8—4881no—
533D8—2441no—
534D8—2442no—
535D8#1175819914yescomplemented
536M16—42no—
537M16—2442no—
538C2×C8—21no—
539C2×C8—611no—
540C2×C8—611no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—4881no—
546C2×C8—4881no—
547C2×C8C2×C22⋊D6111yescharacteristic
548C2×C8C2×C22⋊D6111yescharacteristic
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—41no—
552C4⋊C4—611no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—2441no—
556C4⋊C4D4×D6111yescharacteristic
557C4⋊C4C2×C22⋊D6111yescharacteristic
558C4⋊C4C2×C22⋊D6111yescharacteristic
559C22⋊C4—21no—
560C22⋊C4—21no—
561C22⋊C4—21no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—1221no—
565C22⋊C4—2441no—
566C22⋊C4—2441no—
567C42C23×D6111yescharacteristic
568C23—21no—
569C23—21no—
570C23—21no—
571C23—41no—
572C23—82no—
573C23—82no—
574C23—161no—
575C23—1221no—
576C23—9761no—
577C23—4882no—
578C23—4882no—
579C23—2441no—
580Q8—41no—
581Q8—41no—
582Q8—2441no—
583Q8—2441no—
584D4—21no—
585D4—21no—
586D4—21no—
587D4—21no—
588D4—41no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—81no—
593D4—82no—
594D4—82no—
595D4—82no—
596D4—1221no—
597D4—1221no—
598D4—4881no—
599D4—4882no—
600D4—4882no—
601D4—4882no—
602D4—2441no—
603D4—2441no—
604D4—2441no—
605D4—2441no—
606D4C2×#1175819912yes—
607D4C2×#1175819912yes—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—82no—
619C2×C4—1221no—
620C2×C4—4882no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4C2×D4×D6111yescharacteristic
625C2×C4C22×C22⋊D6111yescharacteristic
626C2×C4#1451982211yescharacteristic
627C8—21no—
628C8—41no—
629C8—1222no—
630C8—1221no—
631C8—1221no—
632C8—2441no—
633C8—2441no—
634C8—2441no—
635C8C2×#1175819912yes—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—164no—
649C22—9764no—
650C22—4881no—
651C22—4881no—
652C22—4881no—
653C22—2441no—
654C22—2441no—
655C22—2441no—
656C22D4×C22⋊D6111yescharacteristic
657C4—21no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—81no—
662C4—4881no—
663C4—2441no—
664C4C22×#1175819911yescharacteristic
665C4#1684055811yescharacteristic
666C4#1684056511yescharacteristic
667C4#1684056611yescharacteristic
668C2—42no—
669C2—42no—
670C2—81no—
671C2—161no—
672C2—9761no—
673C2—4881no—
674C2#2145742411yescharacteristic
675C2D4×#1175819911yescharacteristic
676C2#2145761011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3272C2
socleC22×C61D42
hypercenter#3272D61
hypocenterC61#19884
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3272

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21457632C2semidirect
2. #21457626C2semidirect
3. #21457420C2semidirect
4. #21445409C2semidirect
5. #21445284C2semidirect
6. #21445281C2semidirect
7. #21445258C2semidirect
8. #21445254C2semidirect
9. #21445054C2semidirect
10. #21445052C2semidirect
11. #21444873C2semidirect
12. #21444872C2semidirect
13. #21444702C2semidirect
14. D8×#92418C2semidirect
15. C61×#3272C2semidirect
18. D4×D244C22semidirect
19. #16840165C22semidirect
21. #16840135C22semidirect
22. #16840110C22semidirect
23. #16840108C22semidirect
26. #16840042C22semidirect
27. #16840039C22semidirect
31. #16839967C22semidirect
40. #16839841C22semidirect
41. D4×#92418C22semidirect
42. D4×#92418C22semidirect
45. #16839515C22semidirect
46. #16839514C22semidirect
47. #16839509C22semidirect
48. #16839508C22semidirect
49. #16839506C22semidirect
50. #16839505C22semidirect
51. D4⋊#92418C22semidirect
52. D4⋊#92418C22semidirect
53. D4⋊#92418C22semidirect
54. #16839489C22semidirect
55. C8×#92418C22semidirect
58. D42×C61C22semidirect
59. C61×#492C22semidirect
61. C61×#461C22semidirect
66. C61×#436C22semidirect
188. D8×C61D8semidirect
333. #3272D61semidirect
359. D42C2×D61semidirect
363. #492C2×D61semidirect
375. #461C2×D61semidirect
389. #436C2×D61semidirect
405. C61#19884semidirect
535. D8#11758199semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214576322
#214576262
#214574202
#214454092
#214452842
#214452812
#214452582
#214452542
#214450542
#214450522
#214448732
#214448722
#214447022
D8×#924182
C61×#32722
#1988461